Naturalism and the Apriority of Logic and Arithmetic†
FENG YE*
Apriority traditionally means apriority for a Transcendental Subject. However,
naturalism takes human minds as functions of brains, results of evolution, and
parts of the physical world. There is no Transcendental Subject. Then, is there
still a meaningful conception of apriority? Are logic and arithmetic still a priori?
Moreover, for a naturalist who is also a nominalist, what does our arithmetic
knowledge consist in, if not knowledge about numbers as abstract objects? Is that
knowledge a priori? This paper tries to answer these questions from the
naturalistic and nominalistic point of view. The answers will be subtle.
Naturalism allows us to make some fine distinctions.
1. Introduction
In the previous papers (Ye [2007a], [2007b], [2007c]), I introduced a research project
exploring a truly naturalistic and completely scientific account for human mathematical
practices. It views human mathematical practices as human brains’ cognitive activities
†
The research for this paper is supported by Chinese National Social Science Foundation (grant number
05BZX049). I would like to thank Princeton University and my advisors John P. Burgess and Paul
Benacerraf for the graduate fellowship and all other helps they offered during my graduate studies at
Princeton many years ago. Without them, my researches would not have started.
*
Department of Philosophy, Peking University, Beijing 100871, China.
yefeng@phil.pku.edu.cn, feng.ye@yahoo.com.cn
http://www.phil.pku.edu.cn/cllc/people/fengye/index_en.html
2
and seeks to account for some aspects of human mathematical practices by referring to
the cognitive functions of mathematical concepts and thoughts as mental representations
in brains. Its philosophical basis is naturalism. I take the central thesis of naturalism to be
that human minds are functions of brains, parts of this physical world, and results of
evolution (Ye [2007b]). This is consistent with the so-called ‘philosophical naturalism’ or
‘physicalism’ in the philosophy of mind (Papineau [1993]). It is also consistent with
some of Quine’s characterizations of naturalism, for instance, that humans are physical
denizens of the physical world (Quine [1995]), and that epistemology is a branch of
psychology (Quine [1969]). However, I argued that Quine actually slips away from
naturalism when talking about ‘positing abstract entities’ and I argued that the Quinean
indispensability argument for abstract mathematical entities is based on non-naturalistic
notions (Ye [2007b]). Therefore, this research project also supports nominalism. (See
Shapiro [2005] for surveys on the contemporary philosophy of mathematics.)
One issue that this naturalistic and scientific approach to accounting for human
mathematical practices must deal with is accounting for the apparent universality,
apriority and necessity of logic and arithmetic. Here, there is a challenge for naturalists
and nominalists. Traditionally, apriority means apriority for a Transcendental Subject.
The Transcendental Subject does not belong to the External Reality. Instead, the
Transcendental Subject stands opposite to the External Reality and tries to know things in
the External Reality, either through Senses and Experiences resulted by contacting the
External Reality, or by some innate Reason or Form of Sensibility. Then, something is a
priori for the Subject if the Subject can justifiably know it by innate Reason or Form of
Sensibility alone, without resorting to Senses and Experiences. Logic and arithmetic are
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believed to be a priori in this sense. However, there is no Transcendental Subject for
naturalism. All that exist are brain neurons as the results of evolution and parts of this
physical world, and in causal and other physical connections with other physical things.
Then, is there still an interesting notion of apriority? Are logic and arithmetic still a priori?
Moreover, since numbers as abstract entities do not exist for nominalism, are arithmetic
theorems true after all? If not, then what is arithmetic? What does human arithmetic
knowledge consist in? (Presumably, we do have arithmetic knowledge.) Is that
knowledge a priori as we intuitively hold? How is arithmetic applicable?
Similarly, some philosophers hold that logic and arithmetic truths are analytic,
universal and necessary. What does naturalistic nominalism say about these? Do we still
have meaningful notions of analyticity and necessity under naturalism? Are logic and
arithmetic analytic, universal and necessary?
This paper belongs to the research project. It will look into some of the questions
above. The full naturalistic characterizations of possibility, necessity and analyticity are
big topics and will have to be addressed in other papers. This paper will focus on
apriority and questions related to analyticity, necessity and universality whose answers do
not require the detailed naturalistic characterizations of those notions. I will try to
characterize apriority under the context of naturalism. Then, I will discuss in which
senses logic and arithmetic are or are not a priori, analytic, or necessary. The answers
will be subtle, not the straightforward Yes or No. It seems that the naturalistic framework
allows us to make many fine distinctions. Some of the distinctions will be reminiscent of
the Kantian analytic, synthetic a priori and a posteriori distinctions, but they will be
under the naturalistic setting.
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Some basic assumptions about human cognitive architecture are necessary if one
wants to characterize notions like apriority, analyticity, and necessity under naturalism.
Assumptions taken by this research project are presented in Ye [2007d], [2007e], [2007c].
In the following I will briefly recapitulate the assumptions and introduce some relevant
terminologies.
First, we assume a Representational Theory of Mind. It means that mental
representations (supposedly realized as neural circuitries in a brain) are the building
blocks of human cognitive architecture. The most common types of mental
representations include (nominal) concepts, logical concepts, and thoughts.
Some concepts are Realistic concepts (e.g. DOG). They can represent physical
entities in environments or their properties directly. The semantic mapping rules
determine what they represent. As naturalists, we believe that the semantic mapping rules
can be stated in naturalistic terms (vs. intentional terms such as ‘represent’, ‘mean’, ‘refer
to’ and so on), that is, they can be naturalized (Adams [2003], Ye [2007d]). Some
concepts are cannot represent physical entities, due to their inner structures. They are
called ‘abstract concepts’. Mathematical concepts belong to this category. As nominalists,
we assume that abstract concepts do not represent anything.
A concept has structural constituents. A concept can have a summary feature list
(Murphy [2002], Ye [2007d]), namely, a list of (weighted) other concepts representing
some features of the entities represented by the concept. Concepts are thus mutually
connected by this ‘is a summary feature of’ relation and can even be in circular
connections, but for our ordinary realistic concepts, the semantic mapping rules can still
determine what they represent (Ye [2007d]). Some summary features of a concept can be
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necessary conditions of the concept. For instance, a concept MALE can be a necessary
condition of a concept BACHELOR in someone’s brain. In that case, the semantic
mapping rules determine that whatever represented by BACHELOR are also represented
by MALE. A concept can also have some perceptual memories of object instances
represented by the concept as its structural constituents, called its ‘exemplars’ (Murphy
[2002], Ye [2007d]). For instance, one’s concept RED may contain an exemplar that is a
visual image of a red object. In that case, the semantic mapping rules actually say that an
entity is represented by the concept just in case it has the same color as the exemplar. See
Ye [2007d] for more details.
Thoughts are composed of concepts. Realistic thoughts are composed of realistic
concepts and can represent external physical states of affairs directly. For instance, a
realistic thought <RAVENs are BLACK> in a brain is composed of two realistic
concepts. The thought is true if the physical entities represented by RAVEN have the
physical property represented by BLACK. With the representation relation between
realistic concepts and external physical entities or their properties being naturalized, this
is then a naturalized correspondence theory of truth (for realistic thoughts only).
Moreover, if a concept MALE is a necessary condition of a concept BACHELOR in
someone’s brain, then the semantic mapping rules determine that the thought
<BACHELORs are MALE> must be true. This is an example of analytic thoughts. The
semantic mapping rules guarantee that the naturalized correspondence must exist for a
realistic thought that is analytic. (See Ye [2007e], where the Quinean objections to
analyticity are discussed.) Abstract thoughts in brains, including mathematical thoughts,
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cannot represent external physical states of affairs directly. The naturalized ‘true’
predicate does not apply to them.
Logical concepts can combine other concepts or thoughts to form logically
composite concepts and thoughts. A naturalistic theory of logical concepts is still to be
explored, but we assume that the semantic mapping rules for logical concepts determine
what external affairs correspond to the logically composite thoughts in the expected way.
For instance, the thought ‘p AND q’ corresponds to external affairs that correspond to
both p and q. Then, a logically composite realistic thought that is a logical truth in our
ordinary sense is analytic. That is, the semantic mapping rules for logical concepts
determine that the naturalized correspondence must exist for it.
2. Apriority from the Naturalistic Point of View
Since there are only brains in the natural world and there are no Transcendental Subjects
alienated to nature, apriority can only mean what is a priori for a brain (at a moment).
Therefore, apriority will be a naturalistic notion, a natural property of a thought realized
as neural circuitries in a brain, just like the naturalized ‘true’ predicate or any other
natural properties of physical entities studied by sciences. Note that since the naturalized
‘true’ predicate applies to realistic thoughts only and since apriority implies truth, we will
consider the apriority of realistic thoughts only.
We will make some more assumptions about human cognitive architecture. We
assume that, at each moment, a brain has a conceptual scheme consisting of many
mutually connected concepts. Moreover, a brain has a knowledgebase of factual
knowledge at each moment. It consists of realistic thoughts that are recognized or
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accepted as factually true by the brain, by direct observations, or by accepting what are
learned through linguistic communications and so on. Thoughts in the knowledgebase are
composed of concepts from the conceptual scheme. The conceptual scheme is supposed
to be relatively more stable, while a brain constantly updates its knowledgebase.
Furthermore, we assume that a brain has a collection of thoughts as the brain’s
innate knowledge. They are innate, because they are not obtained by observations, by
learning through linguistic communications, or by inferential processes based on these.
The idea is that human brains have some genetically determined innate capacities, as the
results of evolution. Then, in the individual developmental processes, stimuli from
environments help to make these innate capacities to mature into some practical abilities
and representational knowledge, which can be expressed as the collection of thoughts as
the brain’s innate knowledge. Although stimuli from environments are required for
developing innate knowledge, these thoughts are not bookkeeping records of
observations, or even inductive inferential conclusions from them. They are typically
general knowledge without sufficient inductive bases for an individual. For instance, the
traditional synthetic a priori knowledge in the Kantian sense seems to belong to this
category. In other words, external stimuli required for a brain’s maturation process are
not sufficient if viewed as inductive evidences for these thoughts. External stimuli only
function to help the development of some innate maturation processes, which result in
some very general representational knowledge. Therefore, it is considered innate
knowledge, different from the factual knowledge in the brain’s knowledgebase. (A
couple of examples will be discussed later.)
Then, we can define apriority as follows:
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Definition: A thought is a priori for a brain at a moment, if the brain can
justifiably recognize it as true without resorting to thoughts in the brain’s
knowledgebase of factual knowledge.
I will not try to define ‘justifiably’ in general, which is a different philosophical topic. I
will only rely on presenting specific means by which a brain recognizes an a priori
thought as true and claiming that they are intuitively ‘justifiable’.
First, recall that the semantic mapping rules for realistic concepts determine that a
realistic thought attributing a necessary condition of a realistic concept to the concept (e.g.
<BACHELORs are MALE>) must be true. I will call these ‘simple analytic thoughts’.
Simple analytic thoughts are some examples of a priori thoughts for a brain. We assume
that a brain has the ability to analyze its conceptual scheme, identify necessary conditions
for its concepts, and recognize such simple analytic thoughts as true. This seems to be a
justifiable a priori way to recognize truths. Analyticity is indeed an explanation for
apriority here, as it is envisioned by traditional philosophers such as Frege and Carnap,
but we put it under the naturalistic setting. Other analytic and a priori thoughts include
logical truths. Brains have the ability to analyze the logical structures of logically
composite thoughts and recognize logical truths. This also seems to be a justifiable a
priori way of knowing truths. Logical truths are thus a priori. These two can be
combined. That is, in analyzing the logical structures of logically composite thoughts,
brains can treat components that are simple analytic thoughts as guaranteed truths.
There may be other types of a priori thoughts. A conceivable candidate is a
thought expressed by ‘What is blue is not yellow’. Our color concepts BLUE, YELLOW
and so on may include some visual images of objects with those colors in our memory as
9
constituents, that is, as exemplars of those concepts (Ye [2007d]). Recall that the
semantic mapping rules determine that one’s concept BLUE applies to an object just in
case the object has the same color as the concept’s exemplars. Now, our visual perceptual
subsystem of our cognitive system is such that if one visual mental image is ‘seen’ as
blue, then it won’t be simultaneously ‘seen’ as yellow. This is perhaps determined by the
innate structure of our visual perceptual subsystem, by the way it internally represents
colors. Therefore, perhaps a brain can know that its concepts BLUE and YELLOW are
mutually exclusive by analyzing its visual images as the exemplars of those concepts,
without consulting other factual knowledge in the brain’s knowledgebase. It means that
the flowing thought is perhaps a priori.
（1） <BLUE is-not YELLOW>
Now, this is slightly different from knowing simple analytic thoughts or logical
truths based on identifying the necessary conditions of concepts and analyzing the logical
structures of logically composite thoughts. It involves comparing visual images as
exemplars of concepts. Traditionally, this is considered a resort to intuition, not pure
conceptual analysis. Therefore, whether (1) is analytic may depend on how analyticity is
characterized. In the narrow sense, analytic thoughts include only those that are knowable
by identifying the necessary conditions of concepts and analyzing the logical structures of
logically composite thoughts. Then, (1) is not analytic. In a broader sense, analytic
thoughts may include all those that are knowable by conceptual analyses in a broad sense,
including analyzing perceptual mental images. Then, (1) is analytic. The critical thing
here is to acknowledge that a concept can contain perceptual mental images as
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constituents. I will not here try to explore a full characterization of analyticity under
naturalism. It is another topic. So, I will leave the analyticity of (1) undecided.
A similar problem exists for knowing a little more complex logical truths. A brain
has to rely on manipulating symbols to do logical inferences and this actually
presupposes arithmetic and combinatorial knowledge on counting and concatenating
symbols and so on. When one takes the point of view of a Transcendental Mind, one
easily forgets this fact, for one assumes that the Mind has some mysterious power of
‘grasping’ concepts and thoughts. A brain can only do logical inferences by manipulating
concrete symbols, either within the brain or with the help of hands and papers and pencils.
Therefore, some arithmetic and combinatorial intuition has to be involved. This does
contradict our claim that logical truths are knowable a priori, because manipulating
symbols within a brain is not consulting the factual knowledge in the brain’s knowledge,
although the notion of analyticity has to be carefully characterized in order to include
logical truths.
Another type of a priori knowledge may be human innate knowledge, in the
collection of thoughts as a brain’s innate knowledge mentioned above. For instance,
psychologists discover that infants are already able to distinguish between a white screen
with two black dots and a white screen with three black dots (Lakoff and Núñez [2000], p.
15). After showing an infant a series of screens with two dots in various shapes and in
various positions relative to each other, a screen with three dots is shown, and the infant’s
reaction shows that the infant notices the difference. It means that infants may have the
innate concepts TWO and THREE (where TWO should actually be something like
CONSISTING-OF-TWO-SEPARATE-SALIENT-SMALL-PARTS, and similarly for
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THREE). Moreover, the infant innately treats these as mutually exclusive concepts,
which means that the thought
（2） <TWO is-not THREE>
may be innate knowledge in the infant’s brains.
Such innate knowledge should be a priori, since it does not depend on factual
information received by the brain through sense organs and remembered by the brain in
its knowledgebase. Note that ‘That (pointing to a two-dot-screen) is TWO’ is factual
knowledge recognized by the brain through an observation, but to decide ‘that (pointing
to a three-dot-screen) is not like that (pointing to a two-dot-screen)’, the brain implicitly
resorts to innate knowledge like (2) for distinguishing the two concepts, which is not
itself factual knowledge recognized by the brain through observations.
(2) is similar to (1) but slightly different. Recall that we assume that one’s
concepts BLUE and YELLOW each has a visual image of a blue or yellow object as its
exemplar. Our visual perceptual system can perhaps recognize that whatever that has
exactly the same color as one exemplar will not have the same color as the other. From
there, we can infer that BLUE and YELLOW are mutually exclusive. Therefore, we take
it that (1) is inferred from the structure of the brain’s conceptual scheme alone. As for the
concepts TWO and THREE, we may also assume that one’s innate concepts TWO and
THREE each has a few exemplars that are some visual images of two-dot or three-dot
screens with dots in various shapes and positions. However, these appear insufficient to
infer that nothing will fall under both TWO and THREE. Whether or not something falls
under the concept TWO is still determined by if it is similar to those two-dot screens as
the exemplars of TWO in some relevant aspects, but this ‘in some relevant aspects’ is
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quite complex and is perhaps determined by some innate neural mechanism
implementing a pattern match. We do not have something like a simple color sense to
differentiate between those two-dot screens and three-dot screens. Therefore, (2) is
perhaps not obtainable by analyzing the structures of the concepts alone. A brain brutally
treats its concepts TWO and THREE as mutually exclusive. Therefore, (2) is the brain’s
innate knowledge and it is perhaps not analytic.
Certainly, (2) could be a piece of inductive knowledge. That is, by observing
many instances, one notices that instances belonging to TWO do not belong to THREE.
Then, one infers (2) by an induction. However, the point here is that infants do not seem
to have sufficient observations as the inductive evidences. Moreover, it seems that the
capability of doing inductions is mature much later among children. Therefore, we have
to admit that (2) is innate.
This may be an example of a priori but non-analytic knowledge. All thoughts in
the brain’s collection of thoughts as innate knowledge are similarly a priori but nonanalytic knowledge. Being innate means that they are sort of ‘predetermined harmony’
between brains and environments, as the results of evolution. They are actually some
inductive knowledge of the whole species. But for an individual, they are a priori as we
define it here, since they do not depend on the factual knowledge gathered by an
individual from environments. They are partially genetically inherited. They are typically
general assertions about general regularities of things in environments, such as general
relations between innate concepts, but they are not inductive conclusions based on the
factual knowledge in the knowledgebase, because there are no sufficient inductive
evidences received as factual knowledge by an individual. Moreover, they are typically
13
complex knowledge, which means that they cannot be encoded into the structures of
concepts. (Note that concepts can encode some factual knowledge. See below.) They are
expressed as brutal assertions about some general regularity among physical entities and
their properties in environments, much like (2). Therefore, they are not analytic.
We see that this is reminiscent of the Kantian analytic, synthetic a priori and a
posteriori distinctions, but it is under the naturalistic background. Then, a few fine issues
regarding apriority in the naturalistic setting must be noted.
First, a brain’s conceptual scheme at a moment is a result of the brain’s previous
experiences and it may further develop over time. Therefore, we sometimes carefully say
‘a priori for a brain at a moment’. For example, after learning examples like men legally
married but separated and living alone for many years, and after admitting that they are
also bachelors, one’s concept BACHELOR may undergo some essential change, so that
UNMARRIED is no longer a necessary condition for BACHELOR anymore. Then, the
thought <BACHELORs are UNMARRIED> is no longer analytic. Therefore, analyticity
and apriority are relative to a given conceptual scheme. Note that this is not literally
saying that a previously analytic and a priori thought becomes not analytic or a priori
anymore. Since the necessary conditions of the old concept BACHELOR have changed,
the new concept BACHELOR is actually not the same concept as the old one, although it
is still associated with the word ‘bachelor’ in the brain. Similarly, the new thought is not
the same thought as the old one. A concrete thought does not change.
Second, with this notion of apriority, we can characterize other coarser-grained
notions of apriority that may be of some philosophical interests. For instance, we can
single out some concepts of special interests, for example, logical concepts or arithmetic
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concepts, and then ask if a thought is a priori without considering the internal structures
of other concepts. We can also define a notion of apriority by considering conceptual
schemes in multiple brains, for instance, in all brains of people in a language community.
In that case, we typically use linguistic terms expressing concepts to individuate concepts.
For example, all concepts BACHELOR in all brains (with language competency)
expressed by the same word ‘bachelor’ are consider ‘the same concept’, no matter if they
have UNMARRIED as a necessary condition. Then, we can ask if all thoughts expressed
by ‘bachelors are male’ or ‘bachelors are unmarried’ in all people’s brains are analytic
and a priori. Perhaps, the former thoughts are but the latter are not. This is perhaps what
we ordinarily mean when we say that the public proposition expressed by the sentence
‘bachelors are male’, or the sentence itself, is a priori. Then, for the sentence ‘bachelors
are unmarried’, there is no simple answer for its apriority, in case in half of the brains,
UNMARRIED is still a necessary condition for BACHELOR, and it is not anymore in
the other half of the brains. This and something similar to this are actually Quine’s main
reasons against an analytic-synthetic distinction (Quine [1976], [1980]) and it is actually
a misplaced doubt for the distinction for an individual brain (Ye [2007e]).
Third, since a conceptual scheme is a result of previous experiences, being a
priori does not mean being completely independent of experiences. It is not merely that
experiences may cause a conceptual scheme to change, as the example of BACHELOR
shows. Actually, a conceptual scheme may itself embody some factual knowledge about
the world obtained by observations. For instance, one’s concept RED may include one’s
memory of a visual image of a red object seen before as an exemplar. Having the concept
itself implies having the factual knowledge that something is red. Indeed, this way of
15
depending on experiences is different from the way a factual knowledge in the
knowledgebase depends on experiences. It is the existence of an a priori thought that
depends on the existence of a conceptual scheme, which in turn depends on previous
experiences. Knowing the truth of an a priori thought, given a conceptual scheme, does
not itself depend on extra factual knowledge obtained by experiences.
Similarly, being a priori does not mean being completely independent of any
specific features of the world. The existence of a conceptual scheme depends on some
features of the world, as it is explained above. Moreover, the existence and structure of
human fundamental cognitive architecture also depends on some general features of the
world. It is a result of evolutional selections for brains to fit the features of human
environments. Moreover, human innate knowledge is sort of ‘predetermined harmony’
between human brains and some general features of human environments as the results of
evolution. We will see in the next section that even logical truths depend on some very
general features of humans’ immediate environments.
Furthermore, since a priori truths depend on features of the world, being a priori
does not mean being necessary in its strongest sense, that is, in the sense that necessary
truths do not depend on any specific features of the world. I will give more details on this
in discussing the status of logic and arithmetic in the next two sections.
These should not be surprising from the naturalistic point of view, since brains are
physical entities and results of evolution and individual development. They are born with
innate capacities to develop some structures that will fit some general features of human
environments, and they continuously receive stimuli from environments from the
beginning in their individual developments. Nothing in a brain is completely independent
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of any external stimuli or any features of the world. It will be different if we assume the
point of view of a Transcendental Subject. A Transcendental Subject is viewed as
something not belonging to nature or the External Reality. The Transcendental Subject’s
pure Reason is supposed to be the Inner and can be completely independent of any
features of that External Reality. Therefore, a Transcendental Subject may have a priori
knowledge solely depending on the pure Reason, completely independent of any specific
features of that External Reality, or any experiences that the Transcendental Subject has
from that External Reality. There is no such Inner and External distinction for brains,
which grow out of single cells, controlled by genes and affects by the environments.
Fourth, there may not be a very clear borderline between what are a priori and
what are not. Psychologists agree that there may not be a very clear borderline between
what are innate and what are learned. Typically, brains have some innate capacity, but
learning experiences are required to develop that innate capacity into some practical
ability and to generate some representational knowledge. Again, this is unavoidable from
the naturalistic point of view. However, this vagueness of borderline does not make the
notion of apriority scientifically valueless. Remember that we are pursuing a naturalistic
and scientific description of human cognitive processes, and we are not trying to look for
an a priori metaphysical foundation for human knowledge. As long as there are clear
cases of the a priori and the a posteriori, and as long as the notion does tell a difference
between the ways by which human brains arrive at knowledge in many ordinary cases, it
is a valuable scientific notion. Actually, it seems to be a scientifically important notion in
describing human cognitive processes. From the scientific point of view, the critical
assumption here is that we can analyze the human cognitive architecture as a structure
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consisting of a conceptual scheme, a knowledgebase of factual knowledge, and a
collection of innate knowledge. This is of course only a philosophical assumption and it
must eventually subject to scientific verifications.
Finally, remember that all the above are about realistic concepts and thoughts
only. Since the naturalistic ‘true’ predicate does not apply to abstract thoughts, at least we
cannot say that an abstract thought is true a priori or that it is analytically true. I will
explain what we can say about abstract thoughts later.
3. The Status of Logic
Now, consider if logic is analytic and a priori from the naturalistic point of view, and in
connection with this problem, if logic is necessary or universal.
First, consider logic for realistic concepts and thoughts.
That is, consider a
logically composite thought that is composed of realistic simple thoughts and is an
instance of a logical truth pattern in the ordinary sense. I already hinted that it is analytic
and a priori, because the semantic mapping rules for logical concepts guarantee that it is
true and brains can know this fact by analyzing the logical structure of the thought.
However, we have to make some qualifications. Researches in quantum logic
decades ago suggested that if we interpret the logical constants ‘and’ and ‘or’ used for
composing thoughts representing the states of affairs about electrons in a similar way as
we interpret those constants used for composing thoughts about macroscopic physical
objects, some ordinary logical rule, in particular, the distributive law of ‘and’ over ‘or’,
will not be valid (see, for instance, Hooker [1979]). We do not have to accept the
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quantum logical interpretation of quantum mechanics resulted. This observation is itself
scientifically valid. That is, the truth of the thought expressing the distributive law about
macroscopic objects does depend on some special features of those objects, that is, the
fact that they are stably individualized and deterministic physical objects obeying
classical mechanics laws. Therefore, ordinary logical truths are not absolutely universally
true, in the sense that an instance of a pattern of ordinary logical truth may not be true if
the concepts in the instance represent quantum particles, not medium size classical
objects. At least, the distributive law is not absolutely universal. It is not absolutely
necessary either if we agree that the quantum world is possible.
To see this from another perspective, note that even the emergence of the human
fundamental cognitive architecture and ordinary logical concepts in human brains in
evolution depends on the fact that things in humans’ immediate environments that human
brains and sense organs directly interact with in their evolutional history are stable and
deterministic (relative to human brains). For instance, the summary features of a concept
are supposedly implicitly connected by the logical constant AND. Since the order of
those summary features is perhaps not recorded in brains, this actually presupposes that
the ordinary properties of external things observable by human sense organs and brains
are deterministic and mutually commensurable (in the quantum mechanical sense). That
is, observing one feature does not affect another. Otherwise, brains would not have
developed that fundamental cognitive architecture. Therefore, it should not be surprising
that some logical truth patterns become untrue if instantiated with thoughts about things
other than those that are in humans’ immediate environments and that helped to shape the
human fundamental cognitive architecture in human evolutional history.
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We can still admit that thoughts about medium size physical objects in humans’
immediate environments that are instances of logical truth patterns are a priori. First, we
can agree that concepts are classified into concepts representing things in humans’
immediate environments and scientific concepts representing things far away from
humans, such as micro particles. Second, we can admit that this classification information
for a concept is a part of a conceptual scheme. In fact, a concept representing things in
humans’ immediate environments usually has exemplars, namely, perceptual memories
of instances of the things represented by the concept. Therefore, by analyzing these
exemplars, one will be able to know if the concept belongs to one or the other category.
This is similar to the example (1) in the last section. Then, assuming that brains have the
ability to analyze the logical structures of thoughts, brains will have a priori means to
know logical truths for medium size classical objects.
Moreover, it seems that human logical concepts summarize, relative to human
cognitive architecture, the most general features of medium size physical objects in
humans’ immediate environments, and they are the results of human brains’ adapting to
their immediate environments in evolution. It means that logic may be universal and
necessary in a limited sense. That is, ordinary logical truth patterns are universally true
when instantiated with thoughts about things in humans’ immediate environments, and
for whatever things that human brains and sense organs can rather directly recognize,
ordinary logical truth patterns must be true of them.
Now, consider logic for abstract thoughts. That is, consider a logically composite
thought that is composed of abstract concepts and is an instance of a logical truth pattern
in the ordinary sense. Creating and manipulating abstract concepts and thoughts in brains
20
is like imagining things in brains. It seems that brains follow the same logical rules in
doing inferences on abstract thoughts as the logical rules for realistic thoughts. The
psychological reason for this is obvious. Abstract thoughts have the same structures as
realistic thoughts and the latter are evolutionally more primitive. Humans simply adopt
the logic for the latter when imagining things or developing abstract thoughts. They may
actually employ the same brain resources for reasoning about abstract thoughts, as it is
suggested by some psychologists (e.g. Lakoff and Núñez [2000]).
In developing a mathematical theory, we choose some abstract mathematical
thoughts as axioms and derive theorems. It is sometimes claimed that these theorems are
a priori, but that is no more than saying that ‘Holmes is a detective’ is a priori, simply
because we choose to imagine that way. This notion of apriority is not very interesting.
The cognitive values of abstract thoughts are elsewhere. They are in the applications of
abstract thoughts. In applications, we typically start with realistic premises that are
realistic thoughts about real things in the world, for instance, about the population
growths on the Earth. Then, we translate these realistic premises into abstract thoughts
‘about a mathematical model’, for instance, a differential equation and an initial value for
a differentiable function. We then draw an abstract thought as a conclusion by a
mathematical proof. Finally, we translate the conclusion back into a realistic thought
about populations on the Erath. The cognitive values of abstract thoughts consist in the
fact that they help to finally derive literally true realistic thoughts about real things from
literally true realistic premises about real things. In the model of human cognitive
architecture sketched in the last section, abstract thoughts do not belong to any of the
three components. They are tools to facilitate the processing of concepts and thoughts in
21
those components. An explanation of the applicability of mathematics is supposed to
explain how that is so in a very realistic and scientific manner. See Ye [2007f], [2007g]
for more details. The alleged apriority of abstract thoughts is not relevant here.
On the other hand, it seems that our imagination capabilities are constrained by
the fact that we interact with stable, deterministic, medium-size physical objects only in
our evolutional history and we can directly and clearly perceive stable, medium-size
objects only. As a result, when we imagine things, perhaps we can only imagine them as
something similar to stable, deterministic, medium-size physical objects. Especially,
when we want to imagine things clearly and rigorously, we have to imagine them as
deterministic objects with distinctive identity, much like deterministic objects in the
classical mechanics. For instance, we conceive of a set as a collection of deterministic
elements with clear identity. We cannot clearly imagine microscopic particles with
indeterminate particle identity, for instance. Therefore, we can not directly conceive of a
set with members like these quantum particles. This implies that, in imagining things, if
we want to imagine clearly and rigorously, perhaps we cannot but follow the classical
logic, which summarizes the most general features of the things that we can clearly
conceive of. In fact, when we try to describe the quantum world, we have to do it a little
awkwardly and in a roundabout manner.
This means that the classical logic is perhaps necessary for humans in
manipulating abstract thoughts, or in imagining things. However, note that this is perhaps
limited to imagining finite things. It seems that we do have some latitude of freedom in
imagining infinity, which is not a part of what actually shaped our imagination capability
in evolution. For instance, intuitionists actually suggested another way to imagine infinity.
22
Moreover, as another piece of evidence, recall that when people study multiplevalue logics, they still present their theories using the classical logic, for the sake of
clarity and rigor. Multiple-value logics are perhaps good descriptions of thoughts that
encode, in some simplified manner, information about very complex properties of very
complex things, some of which we cannot clearly conceive of and fully describe.
However, in presenting the theories of multiple-value logics, we want the theories to be
clear and rigorous, so that our assertions can be definite and we can avoid
misunderstandings or confusions among researchers. Therefore, we imaging things that
we can clearly imagine to simulate those multiple truth values in formulating the theories
of multiple-value logics. That is, we still follow the classical logic. This also shows that
the classical may be necessary and universal in the limited sense.
4. The Status of Arithmetic
Finally, consider the same questions for arithmetic. First, we must distinguish between an
abstract thought in arithmetic theory, for example,
（3） ‘5+7=12’,
and the realistic thoughts as the results of applying it, that is,
（4） ‘5-fingers plus 7-fingers are 12-fingers’.
We will consider (3) first.
It is tempting to think that (3) is a priori. This is supported by the following
phenomena. We have
（5） ‘5 liters of alcohol added to 7 liters of water are not 12 liters’,
23
but we do not consider this a counter-example of (3). We only say that (3) is not
applicable in this case. This is sometimes interpreted as meaning that (3) is true by
convention, independent of experiences, and it is thus analytic and a priori. However, if
all applications of (3) were like (5), not (4), then the apriority and/or analyticity of (3)
itself would be pointless. In other words, if it were not for the reason that cases like (4)
have some special epistemic status, whatever epistemic status one assigns to (3) is
meaningless for humans. It would be knowledge about some alleged entities with no
connection with things in the world where we live in.
From the nominalistic and naturalistic point of view, abstract thoughts like
‘5+7=12’ do not directly represent external states of affairs. They have other cognitive
functions in brains. They summarize some factual truths about real things but not some
others. Genuine human arithmetic knowledge consists in knowing that arithmetic
theorems like (3) are derivable from the basic assumptions and calculation rules of
arithmetic and knowing that what are derivable are applicable to some cases as in (4), but
not to some other cases as in (5). This is knowledge about real things, not knowledge
about the alleged abstract entities.
(4) and (5) are instances of the following general format, for a realistic concept C
and for some interpretation ‘plus’ of the arithmetic operation ‘addition’,
（6） 5-Cs ‘plus’ 7-Cs are 12-Cs
Therefore, the more interesting epistemological questions are: Which instances of (6) are
true? What kind of truths are they? Are they a priori, analytic, necessary?
First, we can distinguish the cases where (3) is applied in counting, as in (4), and
the cases where it is applied in calculating the result of measuring some physics
24
quantities, as in (5). People may agree that the latter are a posteriori. Therefore, we will
consider the counting cases.
For (6) to be a case of counting, the concept C there must be a sortal concept (e.g.
APPLE, FINGER), not a material substance concept (e.g. WATER, COPPER). Moreover,
sometimes it may not be very clear if something is countable. For instance, are clouds
countable? How about photons in a light beam, are they countable? Similarly, consider
the particle-count property of a quantum system with micro-particles that have
indeterminate automatic creations and annihilations, or micro-particles without
determinate self-identities. They may be countable, since there is such a particle-count
property. However, arithmetic may not be directly applicable to them. Their particlecount property may be indeterminate and may depend on the observer.
This is related to another problem, the interpretation of ‘plus’ in (6). In a typical
real life scenario of applying (3), we count two piles of objects and then put them
together and count again. The operation ‘put together’ can potentially destroy some of the
objects. We certainly do not want to include such cases as the normal applications of (3).
However, suppose that we first measure the particle-count properties of two physics
systems S1 and S2 of micro-particles, assuming that they are isolated systems, and then we
measure the particle-count property of the composite system S1+S2. Then, we have to
consider potential interactions between the two subsystems, which may affect the
particle-count property of the composite system. One may still retort that arithmetic
applies only to the idealized scenarios where there are no such interactions. However,
everything in this real universe can potentially influence everything else in the universe
(ignoring the limitation due to the finite speed of light signals, gravitational waves and so
25
on). To say that arithmetic applies only to idealized scenarios where there are absolutely
no interactions is to say that arithmetic applies to nothing in this real universe. Moreover,
to say that arithmetic applies only to the scenarios where the interactions are insignificant
and can be ignored is to say that arithmetic applies only to the scenarios in which
arithmetic can be applied. It is circular and it actually does not answer the real question.
Our question is exactly, ‘what is the nature of the knowledge that arithmetic is applicable
to a particular scenario, or what is the nature of the truth that is a result of such an
application?’ Is it a priori that arithmetic is applicable to counting fingers as in (4)?
These observations show that the applicability of arithmetic theorems like (3) to
any real things in this real universe has preconditions. We have at least to see some
fingers in order to now that they are countable. We have to see the kind of operation
involved in counting in order to know that there are no ‘unwanted’ interactions. Even if
one only stands back a step and count two piles of apples together, without moving them,
the fact they there are no ‘wanted’ interactions between the two systems is still due to the
specific features of stable, deterministic medium-size physical objects. If they are not
apples but micro-particles, you have to consider the potential interactions and standing
back a step also changes the reference frame of the observer’s. It means that arithmetic is
not absolutely universally applicable. It depends on some special features of the things to
which it is applied, and the applicability to a particular scenario is not completely
independent of experiences. These should not be surprising. We have seen the same
conclusions for logic.
Now, one may think that there still has to be something that is already in our
minds before we see the fingers or micro-particles and decide if arithmetic is applicable
26
to them, because we have to compare what are seen with what are already in our minds.
What are already in our minds must obey arithmetic laws. Only then can we compare a
real scenario seen with what are already in our minds and decide if arithmetic is
applicable there. That is, one may still think that there still has to be a sense in which
arithmetic is a priori, and is independent of experiences and the features of the world.
The observation is true, but for a naturalist, it is naturally explained in the
naturalistic manner. Physical objects in humans’ immediate environments are
deterministic medium-size objects, with stable and clear identity. Arithmetic summarizes
the most general features of these objects in relevant aspects, regarding the object-count
properties of them. As the results of evolution and individual developments, brains
develop arithmetic concepts for representing these features. When a brain judges that (3)
is applicable in a scenario, it compares the scenario observed with similar scenarios that
are remembered in learning counting. Some innate concepts and knowledge may be
involved in this, for instance, something like the concepts TWO, THREE and (2) in
Section 2, but these are the results of evolution. They are not completely independent of
experiences or the features of the world. If human environments had been such that
nothing was countable there, or nothing had stable, deterministic individual identity there,
and everything there were like clouds or micro-particles with no stable, deterministic
self-identity, then evolution would not have selected the human conceptual scheme with
the number concepts as they have now.
Moreover, it seems that human cognitive architecture is shaped by evolution in
such a way that humans can directly and clearly perceive only stable, deterministic
physical objects for which arithmetic is applicable, and similarly, when brains imagine
27
individual things, they can only imagine them as something similar to stable,
deterministic physical objects in humans’ immediate environments. That is, similar to
logic again, arithmetic may be necessary for humans in a limited sense.
Now, come back to the question of apriority of (6). We agree that for this
scientific notion of apriority, what is a priori needs not be absolutely universal, or
completely independent of experiences or the features of the world, and we agree that
logic is still a priori in this scientific sense. How about arithmetic or the instances of (6)?
First, is the knowledge that FINGER is a sortal concept, or that fingers are
countable, some a priori knowledge? Or is it some factual knowledge in a brain’s
knowledgebase? If it is the latter, then (4) has to be a posteriori, because one must know
that fingers are countable before one knows that arithmetic applies to fingers.
There are reasons to classify such sortal vs. non-sortal distinction about concepts
as a part of a conceptual scheme. Suppose that a concept WATER or FINGER in a brain
has exemplars. Then, by analyzing the perceptual images in those exemplars, perhaps the
brain is already able to decide if the concept is a sortal concept. (Perhaps some general
innate knowledge about what sort of things are countable is also involved.) This does not
make the judgment that a concept is a sortal concept completely independent of
experiences. One at least has to see actual fingers in order to know that they are countable.
However, like the example (1) in Section 2, the way it depends on experiences is through
the formation of a conceptual scheme. It does not depend on extra factual knowledge
registered by the brain based on the conceptual scheme. In that sense, we may want to
consider the knowledge of a concept’s being a sortal concept as a priori knowledge,
obtainable by analyzing the conceptual scheme alone, without consulting the factual
28
knowledge expressed with that conceptual scheme. Remember that we already agree that
being a priori does not mean being completely independent of experiences.
Second, recall that for (6) to be true, besides the requirement that C has to be a
sortal concept, the operation ‘plus’ there must also satisfy some condition, that is, it
should not cause ‘unwanted’ interactions regarding the object-count property. Let us
assume that we have a concept, ADDITIVE, representing operations satisfying the
condition. Then, is it a priori knowledge that a particular instance of ‘plus’ (e.g. putting
fingers together and counting) is ADDITIVE? Perhaps we can answer ‘Yes’ for the same
reason again. An exemplar of a kind of operations may be sufficient to reveal that the
kind of operations is represented by our concept ADDITIVE.
So, we consider this conditional statement:
（7） If C is a sortal concept and ‘plus’ is additive, then 5-Cs ‘plus’ 7-Cs are 12-Cs.
Is it a priori?
Here, we finally get to the point where we have to explain what number concepts
are, and what structures they have, and how they are related to each other. This is a big
topic in itself. Here I can only offer some preliminary analyses. It seems that people do
not have number concepts ‘1’, ‘2’, ‘3’, and so on singly. Instead, a brain memorizes
numerals ‘1’, ‘2’, ‘3’ and so on in a numeral system, and memorizes the associations
between these numerals and the body’ motor actions in counting. These include the rules
for carrying digits when counting over 9. These memories are the constituents of one’s
number concepts 1, 2, 3 and so on as a whole. With these memories, a brain can control
the body’s counting motor actions. A specific number concept, say 7, has the further
function of controlling the termination of a series of counting motor actions, namely,
29
counting up to 7. Note that a brain can count its own counting motor actions, or count
memories of numerals in the brain. That is, a brain’s counting actions do not require that
there are physical objects out of the brain to be counted. Counting concrete physical
objects out of the brain may be required in learning counting and in developing the
number concepts in the brain. The point here is that, after the conceptual scheme with
number concepts is formed, the brain does not need to go out of itself to count things.
Moreover, the brain has a standard interpretation of ‘plus’, that is, something like
‘counting two sequences separately and then count through them together’. With that, a
brain can derive
（8） ‘5-counts added to 7-counts are 12-counts’
by following those counting rules and doing the counting in the brain, without the need to
consult the factual knowledge in the brain’s knowledgebase. Therefore, we can perhaps
agree (8) is a priori. It is about the result of a brain’s inner counting actions in following
those counting rules. Here, I use ‘added to’ to mean the brain’s inner action of ‘counting
through together’, that is, the standard interpretation of ‘plus’ for counting actions in the
brain. Note again that this does not mean that (8) is absolutely necessary, or completely
independent of any features of the world. (8) actually states some natural regularity
among brains’ inner cognitive activities, and it is a priori because a brain can know it by
performing those activities inside the brain by itself.
Now, to go from (8) to (7), we still need something else, because (7) is about
physical entities out of the brain, not about the brain’s inner counting actions. First, we
must explain what the composite concept ‘5-Cs’ represents when C is a sortal concept
representing physical entities out of the brain. Here, we assume that the semantic rule for
30
the composite concept ‘5-Cs’ determines that a collection of physical objects is
represented by the composite concept, if the objects in the collection are represented by
the concept C, and there is a one-one correspondence between the physical objects in the
collection and the brain’s counting actions for counting up to ‘5’.
This alone is not sufficient to derive (7) from (8) because we still need something
to bridge between the ‘plus’ operation in (7) and the ‘added to’ operation in the brain. For
example, in the simplest case, one counts two piles of pebbles on a table separately, and
then one stands a step back and counts them together. This ‘standing a step back and
counting them together’, though simple enough, is still not the same as ‘counting through
together’ within a brain in the brain’s inner counting actions. In particular, consider the
fact that there are non-additive operations and there are the borderline cases between the
additive and the non-additive. The correspondence between counting one’s own inner
counting actions and counting some external physical objects in a real life situation is not
absolutely and universally guaranteed.
It seems that the bridge should be something like the following:
（9） If C is a sortal concept and ‘plus’ is additive, then 5-Cs ‘plus’ 7-Cs are 12-Cs, if
and only if 5-counts added to 7-counts are 12-counts.
This actually says that there is a correspondence between an additive operation ‘plus’ in
counting countable physical entities out of a brain and the operation ‘added to’ for the
brain’s own counting actions. It seems that failing to notice this bridge invites people to
conclude hastily that (6) is a priori.
One might insist that arithmetic means only (8), not (6). However, similar to the
situation between (3) and (4), if (6) had not any special epistemic status, the apriority or
31
analyticity of (8) alone would not tell much. Actually, if (6) had not been the case,
probably human brains would not have developed the conceptual scheme of number
concepts as they have now. The status of (6) or (7) is still an interesting question.
Then, what is the epistemic status of (9)? First, it seems that (9) cannot be analytic
even in the broader sense of analyticity. Consider the fact that some operations of ‘adding
together and counting through’ are not additive. If (9) is to be analytic, it has to be due to
some definitive conceptual structure of the concept ADDITIVE. The concept ADDITIVE
may contain some general features and/or exemplars of the operations that are ‘additive’,
that is, that are similar to the sort of operations that do correspond to the ‘added to’
operation for counting actions within the brain. However, this seems insufficient to imply
a general correspondence between the operation on external physical entities and the
operation on the brain’s counting actions. What (9) states is a very general natural
regularity among natural phenomena, namely, between some processes in brains and
some processes among other physical objects. It seems that such a general regularity
about external physical entities cannot be derived from the conceptual features or
exemplars of the concept ADDITIVE by conceptual analyses alone.
Note that we should not define ADDITIVE by the condition ‘whatever that do
correspond to the “added to” operation for counting actions within a brain’. This will
make (7) trivial, for it then actually says that if the operation ‘plus’ does satisfy the
addition rules used by a brain then 5-Cs ‘plus’ 7-Cs are 12-Cs. Moreover, the condition
itself must be verified empirically, by actually counting external physical entities,
performing the operation ‘plus’ on them and then counting again. Then, whether or not
the ‘plus’ operation in counting fingers is an additive operation will not be a priori. Here,
32
one only shifts the critical point from a bridge like (9) to a condition on what is to be
ADDITIVE. Therefore, it still cannot imply that (6) is a priori.
On the other side, I admit that it is not completely clear if a brain’s knowledge
similar to (9) is the result of some inductive reasoning based on experiences or it is innate.
It seems that the number concepts for some very small numbers may be innate. For
instance, the concepts TWO, THREE discussed in Section 2 may be innate. Then, there is
also some a priori innate knowledge involving these concepts. However, for our more
complex number concepts, the facts seem to be that a large amount of experiences are
involved in learning counting and in developing our conceptual scheme of number
concepts, and these experiences also produce knowledge like (9). That is, the experiences
of learning counting various kinds of physical objects also inductively support the claim
that there are correspondences between the ordinary operations ‘plus’ in counting some
types of physical objects and the operation ‘added to’ in our inner counting actions. The
experiences also help to develop the brain’s ability to recognize which operations on
external physical entities are additive and which are not. Certainly, some innate human
brain capacity is the precondition for the fact that humans can learn counting and can
develop number concepts. However, the necessary large amount of learning experiences
seems to indicate that (9) is closer to a generalization from experiences. That is, in
counting pebbles in learning counting, one recognizes that there is a correspondence
between the ‘added to’ operation on one’s own counting actions in the brain and the
‘gathering together and counting again’ operation on the pebbles. Then, one generalizes
this to other countable physical objects and other ‘plus’ operations on countable objects.
33
One might insist that the empirical evidences are not sufficient to support the
universal generality of arithmetic. However, we have agreed that arithmetic is not really
absolutely and unconditionally universal. The jump from recognizing (9) for pebbles to
hypothesizing it for all other medium-size, deterministic physical objects is no bolder
than the jump from seeing apples falling down the tree to the hypothesis that the same
gravitational force explains the motions of the Moon around the Erath.
Therefore, I am more inclined to say that (9) is a posteriori. Note that this does
not contradict the claim that (6) has some special status for humans. Recall that we
already agree that arithmetic laws summarize the most general features of physical
objects in humans’ immediate environments and that arithmetic is necessary is a limited
sense, that is, in the sense that human imagination capability is constrained by the
evolutional environments for human brains so that humans can clearly conceive of only
entities similar to those physical objects in humans’ immediate environments. That is
how (6) differs from the Newtonian gravitational law. Applications of arithmetic like (6)
seem to belong to the cases where human innate cognitive architecture already
determines that, within some limited scope, things cannot be otherwise, for otherwise that
innate cognitive architecture would not have evolved in evolution. However, at the same
time, a large amount of learning experiences and some generalizations from experiences
are still necessary for an individual brain to reach the relevant general knowledge, and
therefore, we cannot simply count all the relevant knowledge as innate for an individual
brain.
34
Finally, I will briefly examine what is perhaps the most common account for (3)
and its applications in contemporary logic and philosophy. It is alleged that (3) actually
means something like
（10）5xPx7yQyz(PzQz) 12z(PzQz)
where 5xPx is the first order formula saying that there are exactly 5 individuals that are
P. Spelled out, it will contain 5 existential quantifiers and inequalities. (10) is a logical
truth in the first order logic. Therefore, it is alleged that (3) is also a logical truth, and
therefore it is a priori and its applicability to anything is guaranteed. Sometimes, it is
further alleged that (10) is analytic, knowable by conceptual analyses alone, without
resorting to intuitions. See, for example, Yablo [2002] and similar accounts in Hofweber
[2005]. Frege’s [1884] account is also similar, although it is couched in the second order
language (see also Demopoulos [1997]).
Considering examples such as counting water, clouds, or micro-particles, we will
naturally suspect that there must be something missing in this account. First of all, it is
not about the actual number concepts in most human brains. It is about some logicians’
inventions. This may have value for a Transcendental Mind, for the Mind wants to make
sure that His or Her arithmetic knowledge is a priori and universally applicable. This
paraphrasing might help for that. However, its value for a scientific description of how
brains actually work is not clear.
Secondly, this paraphrasing does not really show that arithmetic is universally,
unconditionally applicable. The concepts P and Q in (10) must be sortal concepts, and we
must at least look at some samples of water and fingers in order to know that water is not
countable and fingers are. Moreover, the logical disjunction in (10) is interpreted as the
35
operation ‘plus’ in (6) in applying (10) to get (6). This requires the precondition that the
operation ‘plus’ does follow the logical rules for disjunction. We know that this might not
be the case for quantum particles. Furthermore, for the logical inferences on formulas like
(10) to correctly represent counting some physical objects, there must be some sort of
correspondence between counting the physical objects and manipulating symbols in
doing logical inferences on formulas like (10). That is, something similar to (9) is still
necessary. It will express the general natural regularity in the correspondences between
the counting pebbles and the symbolic manipulations on formulas like (10). More
specifically, when you manipulate those 5 quantifiers in 5xPx one by one, it must
correspond to counting those 5 pebbles one by one. Similarly, when you gather those
quantifiers in 5xPx and in 7xPx, and arrange them into a sequence of 12 quantifiers in
12xPx, it must correspond to the same operations on the two piles of pebbles. These are
the preconditions for applying (10) to get something like (6). Therefore, paraphrasing (3)
into (10) does not straightforwardly show that (3) is unconditionally applicable or that (6)
is a priori.
Finally, obviously, counting actions are involved in manipulating formulas like
(10), either within a brain or by utilizing papers and pencils. One has to count the number
of quantifiers to make sure that they are correct. If the knowledge about counting is based
on intuition, not conceptual analysis alone, then doing logical inferences on (10) already
resorts to intuitions. I have mentioned this point in Section 2. It means that paraphrasing
(3) into (10) cannot prove that arithmetic knowledge does not involve any intuitions.
We see that none of the fine issues we discussed above can be really bypassed by
paraphrasing (3) into (10).
36
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